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Manuscript Title: Generation of molecular symmetry orbitals for the point and double groups
Authors: K. Rykhlinskaya, S. Fritzsche
Program title: BETHE
Catalogue identifier: ADUH_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 171(2005)119
Programming language: MAPLE 7 and 8.
Computer: All computers with a license for the computer algebra package MAPLE [1].
Operating system: Linux 8.1+ and Windows2000.
RAM: 10-30 MB
Keywords: atomic and molecular orbital, double group, point group, projection operator, symmetry orbital.
PACS: 02.20.-a., 31.15.Hz.
Classification: 16.1, 16.3, 14.

Nature of problem:
Molecular and solid-state quantum computations can be simplified considerably if the symmetry of the systems with respect to the rotation and inversion of the coordinates is taken into account. To exploit such symmetries, however, symmetry-adapted basis functions need to be constructed instead of using -- as usual -- the atomic orbitals as the (one-particle) basis. These so-called symmetry orbitals are invariant with respect to the symmetry operations of the group and are different for the point and double groups, i.e. for nonrelativistic and relativistic computations.

Solution method:
Projection operator techniques are applied to generate the symmetry-adapted orbital functions as a linear combination of atomic orbitals.

Reasons for new version:
Inclusion of new procedures to generate symmetry orbitals.

Summary of revisions:
The following procedures have been added or amended.
  1. AO() - Auxiliary procedure to represent an atomic orbital (r | a nlm) which is centered at the position a = (a1, a2,a3).
  2. SO() - Auxiliary procedure to represent a symmetry orbital (r | (Ga) nlm; Τ(α)μν).
  3. Abasis() - Auxiliary procedure to represent an atomic basis set {(r | a nlm)} which is centered at the position a = (a1, a2,a3).
  4. Bethe_generate_AO() - Generates a list of atomic orbitals (including all m's) at the site a = (a1, a2,a3) and for an atom with the identifier stringatom.
  5. Bethe_generate_AO_basis() - Generates an atomic basis by applying all symmetry operations of the point group ς with label Glabel to the atomic orbitals AO1, AO2, ... of a given orbital basis.
  6. Bethe_generate_SO() - Expands a symmetry orbital (r | (Ga) nlm; Τ(α)μν) in terms of the atomic orbitals of a set of equivalent atoms.
  7. Bethe_generate_SO_basis() - Generates a complete, but linear independent basis of symmetry orbitals for the point group ς with label Glabel from the set of atomic orbitals as described by the atomic basis sets Abasis1, Abasis2, ... .
  8. Bethe_group() - Provides the basic group data and notations.
  9. Bethe_set() - Defines either a relativistic or nonrelativistic framework for the generation of the atomic orbitals and the internal interpretation of the quantum numbers.

Restrictions:
The generation of the symmetry orbitals is supported for the cyclic and related groups Ci,Cs,Cn,Cnh ,Cnv, the dihedral groups Dn,Dnh, Dnd, the improper cyclic groups S2n (n <= 10), the cubic groups O, T, Oh, Th, Td as well as the icosahedral groups I and Ih. In all these cases, the symmetry orbitals can be obtained for either the point or double groups by using a nonrelativistic or, respectively, relativistic framework for the computations.

Unusual features:
All commands of the BETHE program are available for interactive work. Apart from the symmetry orbitals generation, the program also provides a simple access to the group theoretical data for the presently implemented groups from above. The notation of the symmetry operations and the irreducible representations follows the compilation by Altmann and Herzig [2]. For a quick reference to the program, a description of all user-relevant commands is given in the (user) manual Bethe-commands.ps which is distributed together with the code.

Running time:
Although the program replies 'promptly' on most requests, the running time depends strongly on the particular task.

References:
[1] Maple is a registered trademark of Waterloo Maple Inc.
[2] S. Altmann and P. Herzig, Point-Group Theory Tables (Clarendon Press, Oxford, 1994).